Dice in Line

AtCoder

IDabc154_d

Time2000ms

Memory256MB

Difficulty—

We have $N$ dice arranged in a line from left to right. The $i$\-th die from the left shows $p_i$ numbers from $1$ to $p_i$ with equal probability when thrown. We will choose $K$ adjacent dice, throw each of them independently, and compute the sum of the numbers shown. Find the maximum possible value of the expected value of this sum. ## Constraints * $1 ≤ K ≤ N ≤ 200000$ * $1 ≤ p_i ≤ 1000$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $K$ $p_1$ ... $p_N$ [samples]

Samples

Input #1

5 3
1 2 2 4 5

Output #1

7.000000000000

When we throw the third, fourth, and fifth dice from the left, the expected value of the sum of the numbers shown is $7$. This is the maximum value we can achieve.

Input #2

4 1
6 6 6 6

Output #2

3.500000000000

Regardless of which die we choose, the expected value of the number shown is $3.5$.

Input #3

10 4
17 13 13 12 15 20 10 13 17 11

Output #3

32.000000000000

API Response (JSON)

{
  "problem": {
    "name": "Dice in Line",
    "description": {
      "content": "We have $N$ dice arranged in a line from left to right. The $i$\\-th die from the left shows $p_i$ numbers from $1$ to $p_i$ with equal probability when thrown. We will choose $K$ adjacent dice, throw ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc154_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have $N$ dice arranged in a line from left to right. The $i$\\-th die from the left shows $p_i$ numbers from $1$ to $p_i$ with equal probability when thrown.\nWe will choose $K$ adjacent dice, throw ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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